How can I prove Jerusalem was destroyed in 586 BC and not 607 BC?

I doubt that it can be proved conclusively because we have to go outside of the Bible for secular dates. Secular dates are not something that I have any confidence in. I can only acquaint you with the work of Dr Floyd Nolen Jones which I reproduce below:

The Year Nebuchadnezzar Razed Jerusalem

[Chrono­logical Compendium from page xiii.]

Genesis

Event

Yrs.

AM

Age of Earth

Genesis

Event

Yrs.

AM Age of Earth

Ch. 1

Creation

0

0

7:6

The Flood when Noah was 600

1656

5:3

Seth born when Adam was

130

130

11:10

Arphaxad born when Shem was

100

1658

5:6

Enos born when Seth was

105

235

11:12

Salah born when Arphaxad was

35

1693

5:9

Cainan born when Enos was

90

325

11:14

Eber born when Salah was

30

1723

5:12

Mahalaleel born when Cainan was

70

395

11:16

Peleg born when Eber was

34

1757

5:15

Jared born when Mahalaleel was

65

460

11:18

Reu born when Peleg was

30

1787

5:18

Enoch born when Jared was

162

622

11:20

Serug born when Reu was

32

1819

5:21

Methuselah born when Enoch was

65

687

11:22

Nahor born when Serug was

30

1849

5:25

Lamech born when Methuselah was

187

874

11:24

Terah born when Nahor was

29

1878

5:28

Noah born when Lamech was

182

1056

11:26

Abraham born when Terah was

130

2008

11:10

Shem born when Noah was

502

1558

12:4

Abraham enters Canaan, age 75

75

2083

Scripture has several large time spans that enable us to begin at 2083 AM & quickly obtain a BC date for Creation.

Scripture

Event

Yrs

AM Age of Earth

Gen. 12:4

Abraham enters Canaan and begins sojourn, age 75

75

2083

Gen 12:10; Ex 12:40; Gal 3:17

From when Abraham left Haran to enter Canaan until the Exodus from Egypt (to the very day)

430

2513

1 Kings 6:1

Exodus to start of Temple, 479 years (in the 480th year which is 479 years plus 16 days...)

479

2992

1 Kings 11:42; 6:1,37-38

Start of Temple to division of the Kingdom. Solomon reigned 40 yrs, Temple begun in his 4th year

37

3029

Ezek 4:4-6

Division of kingdom to destruction of Jerusalem in the 390th year (inclusively numbered = 389+)

389

3418

Without the use of any resources other than the Authorized Bible, the date Nebuchadnezzar burned Jerusalem and the Temple was...derived... by simply utilizing all the Scriptures exactly as recorded in the Chrono¬logical Compendium on page xiii. [Tables above] As already documented in our second chapter, these few passages established the year of the fall of the Kingdom of Judah in the 11th year of Zedekiah as AM 3418. The relevant Scriptures relating to the reigns of the kings of Judah were then charted in terms of AM years. This resulted in fixing the year of "the Captivity" as AM 3407.

Inclusively numbering 37 years from this first year of Jeconiah's (Jehoiachin) Captivity brought me to 3443 AM, the first year of the reign of Evil-merodach, son and successor of Nebuchadnezzar (2 Kings 25:27, cp. 2 Kings 24:12, 15). Counting back from 3443 AM, 3418 was found to be the 19th year of Nebuchadnezzar - the very year the Scriptures say he razed Jerusalem (Jer. 52:12-15). We have just seen in the above paragraph that this destruction took place in the 11th year of Zedekiah (2 Kings 25:1-7), thus confirming the synchronization of Judah and Babylon in 3418.

Numbering from Nebuchadnezzar's 19th year set AM 3399 as the year he ascended the throne. The Bible tallied that date with the 4th year of Jehoiakim (Jer. 25:1, cp. 46:2). The reader will note that connecting these two kingdoms was accomplished using only Scripture. No secular data whatsoever was consulted; neither was any needed.

With the kingdoms of Judah and Babylon thus firmly synchronized, all that remained was to convert the Anno Mundi years to BC. Of course, as "BC" is a secular designation, consulting such data will be necessary for this step.

The simplest, most accurate and most direct way to accomplish this is to begin with the year of Nebuchadnezzar's accession to the throne of Babylon. It is the most reliable point of contact between the secular kingdoms and that of the Hebrew for the chronologer to so convert. As stated earlier in this treatise, Nebuchadnezzar's accession year is fixed by the well-documented lunar eclipse recorded by Ptolemy. This eclipse took place during the fifth year of Nebuchad¬nezzar's father, Nabopolassar. Again, it has been dated as 22 April, -620 (Julian Period, the historical Gregorian date is April 15, 621 BC).

The Canon of Ptolemy and the Babylonian Chronicles tell us that Nabopolassar, reigned twenty-one years. Hence, if we take 621 BC as the fifth year of his reign, the year of his death and the accession of his son will be fixed at 605 BC. Again, Jeremiah 25:1 states that Nebu¬chadnezzar's first year was the fourth year of the reign of Jehoiakim and this determines the BC date for that Judaic sovereign. Jeremiah also records that the Temple was burned in the nineteenth year of Nebuchadnezzar (Jeremiah 52:12-14, also 2 Kings 25:8); thus, the date for that conflagration is established as 586 BC.

Extract from Page 309:

The Chronology of the Old Testament Dr. Floyd Nolen Jones 16th Edition.

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